108 G. W. Littlehales — Isolated Shoals in the Open Sea. 



X 



sm -\ = 1 



m 



tn—x 

 or s = — r - 



m 



And since x can only vary between zero and m, the proba- 

 bility of having an error between x and x+Jx 



p 1 = — — . dx (1) 



The causes which produce the grouping of a number of 

 deduced geographical positions around the true one are of two 

 kinds ; one tending to place the deduced latitude to the north 

 or south of the true latitude, and the other tending to place 

 the deduced longitude to the east or west of the true longi- 

 tude. So that a particular deduced geographical position P 

 will be the result of having an error OA in latitude and an 

 error OB in longitude. 



2. 



Y 

 A -P 



' ' B~ 



The probability that the geographical position deduced by 

 A, upon his discovery of the shoal, occupies a certain position 

 with reference to the true geographical position of the shoal is, 

 therefore, easily deduced. Through the true geographical 

 position of the shoal let two rectangular axes, OX and OY, be 

 passed as shown in figure 2. Upon the former conceive errors 

 in longitude to be measured, and upon the latter, errors in 

 latitude. The position P, of which the coordinates are x and 

 y, results from the concurrence of two conditions, the error of 

 x miles in longitude and the error of y miles in latitude. The 

 probability p t of the error x is, as shown by equation (1), 



m— x 



p^ = — s— . dx 



m 



and, in the same manner, the probability jp 3 of the error y will be 



P^-^r-dy (2) 



In these formulas, m and n represent respectively the 

 extreme errors in longitude and latitude in miles. 



